However, typically functors of interest do not respect weak equivalences and hence do not uniquely or even naturally give rise to an (∞,1)-functor. In general, they contain too little information to accomplish this. Notably, to objects x,y∈Cx, y \in C that are equivalent in C\mathbf{C} but not isomorphic in CC, the functor will in general not assign objects F(x)F(x) and F(y)F(y) that are equivalent in D\mathbf{D}, as an (∞,1)-functor would. So it matters on which representatives of a C\mathbf{C}-equivalence class of objects the functor FF is applied.

Remembering that by Dwyer-Kan simplicial localization the morphisms in C\mathbf{C} and D\mathbf{D} are zig-zags of morphisms in CC and DD, a very general notion of derived functor therefore takes a derived functor of FF to be a functor 𝔻F:C→D\mathbb{D}F : \mathbf{C} \to \mathbf{D} induced from the universal property of the localization by a functor of the form F∘Q:C→DF \circ Q : C \to D, where Q:C→CQ : C \to C is an endofunctor which is naturally connected to the identity by a zig-zag of weak equivalences:

Here if this zig-zag consists just of one morphism to the left one would speak of a left derived functor. If it consists of just one morphism to the right, one would speak of a right derived functor. In general, it is just a derived functor.

On model categories

In highly structured situations where CC and DD are equipped not just with weak equivalences but with the full structure of a model category and if FF is a left or right Quillen functor with respect to these model structures, there are accordingly more structured ways to solve this problem:

where QC:C⟶Ho(C)Q_C \colon C \longrightarrow Ho(C) is the universal morphism characterizing the homotopy category and similarly for QDQ_D.

There is a general method of ordinary category theory to solve such problems universally: one may take Ho(C)→Ho(D)Ho(C) \to Ho(D) to be either the left or right Kan extension of Qd∘FQ_d \circ F along QCQ_C.

In the literature this is often takes as the definition of total left or right derived functors. Unfortunately, it is not clear how this definition by Kan extension relates to what should be the right (∞,1)-category theoretic situation above. Moreover, the examples of derived functors that play a role in practice are effectively always constructed instead rather by combining FF with cofibrant/fibrant or similar replacement functors. It is then but a happy byproduct that the functors so obtained also happen to be left or right Kan extensions.

(if it exists). Dually, the right derived functorRFR F of FF is its left Kan extension along pp. Note the reversal of handedness; this is unfortunate but unavoidable.

More generally, if DD is itself a category with weak equivalences, then by derived functors of FF we often mean derived functors of the composite

C→FD→HoD C \stackrel{F}{\to} D \to Ho_D

Remark

By the universal property of HoCHo_C, functors HoC→DHo_C \to D are equivalent to functors C→DC\to D which take weak equivalences to isomorphisms. If FF itself takes weak equivalences to isomorphisms, then its left and right derived functors are both (isomorphic to) its unique extension along pp. In general, however, LFL F and RFR F are not extensions of FF even up to isomorphism.

Remark

Remark

If the codomain admits sufficiently many limits and colimits, a Kan extension can be computed in terms of those, and that such Kan extensions are called pointwise. Homotopy categories generally do not admit even small limits and colimits, and moreover the domains of the functors in question are generally large, so such a construction of a derived functor is not possible.

However, when derived functors are constructed using fibrant and cofibrant replacements, as above, it turns out a posteriori that they are actually pointwise: they are preserved by all representable functors, and hence their individual object values have the universal property of the (generally large) limits that would have been used to compute them, even though not all limits exist in the homotopy category. In fact, derived functors constructed in this way are actually absolute Kan extensions: preserved by any functor whatsoever.

By taking quasi-isomorphisms as weak equivalences, Ch•(𝒜)Ch_\bullet(\mathcal{A}) is naturally a category with weak equivalences. In much of the literature on homological algebra, the refinement of this structure to a projective or injective model structure on chain complexes is implicit. For instance, an injective resolution of chain complexes is nothing but a fibrant replacement in the injective model structure. Dually, a projective resolution is a cofibrant replacement in the projective model structure. (Note, though, that hypotheses on 𝒜\mathcal{A} are required in order for these model structures to exist.)

Note first that Ch•(F)Ch_\bullet(F) automatically preserves chain homotopies, and therefore also preserves chain homotopy equivalences. Since the projective (resp. injective) model structure on chain complexes has the property that weak equivalences (that is, quasi-isomorphisms) between cofibrant (resp. fibrant) objects are chain homotopy equivalences, it follows that Ch•(F)Ch_\bullet(F) automatically preserves weak equivalences between projective-cofibrant objects, and also between injective-fibrant objects. Thus, it has a left derived functor if the projective model structure on Ch•(𝒜)Ch_\bullet(\mathcal{A}) exists, and a right derived functor if the injective model structure exists.

In the homological algebra literature, what is called the ppth right derived functor

The last morphism computes the cochain cohomology of the resulting cochain complex in degree 0.

Of course, it is equivalent to instead regard AA as concentrated in degree 00, and then take the ppth homology group at the last step. Left derived functors are dual, using the projective model structure.

The first and the last steps are traditionally included, but are not really necessary:

Instead of applying the first step and restricting attention to arguments that are chain complexes concentrated in a single degree, one can evaluate ℝCh•(F)\mathbb{R} Ch_\bullet(F) on all chain complexes (and then, if desired, take homology groups). In homological algebra one then speaks of hyper-derived functors.

The last step of taking cohomology groups serves to extract invariant and computable information. It also destroys the simple composition law of functors, though. But there is a computational tool that can be used to recover the derived functor – in this homological sense – of the composite of two functors from their individual derivations: this is the spectral sequence called the Grothendieck spectral sequence.

Long exact sequences

Traditionally, in homological algebra, one only takes left derived functors of right exact functors, and right derived functors of left exact ones. As we saw above, both left and right derived functors can be defined without these hypotheses, but it is only in the presence of these hypotheses that we obtain long exact sequences.

is a short exact sequence of chain complexes. But since QCQ C is projective, this short exact sequence is split, and therefore preserved by any additive functor. Thus we have another short exact sequence

This is how derived functors are traditionally introduced in homological algebra: as a way to continue the right half of a short exact sequence preserved by a right exact functor into a long exact sequence. The case of left exact functors and right derived functors is dual.

Examples

The (total) derived functor of the limit functor is the homotopy limit. The functors lim(i)lim^{(i)} often called the derived functors of Lim are then given by the (co)homology of that ‘total’ form.

Functoriality

Passage to left derived functors is a pseudofunctor from a 2-category of model categories, left Quillen functors, and natural transformations to Cat, and similarly for right derived functors. These can be combined into a double pseudofunctor? from the double categoryof model categories? to the double category of quintets in Cat, which implies that some mates are also preserved by deriving, even when they relate composites of left and right Quillen functors; see (Shulman).